Question: Simplify and expand the following expression: $ \dfrac{4r - 10}{r + 8}-\dfrac{4r}{r - 10} $
Answer: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(r + 8)(r - 10)$ Multiply the first term by $\dfrac{r - 10}{r - 10}$ $ \begin{align*} \dfrac{4r - 10}{r + 8} \times \dfrac{r - 10}{r - 10} & = \dfrac{(4r - 10)(r - 10)}{(r + 8)(r - 10)} \\ & = \dfrac{4r^2 - 50r + 100}{(r + 8)(r - 10)}\end{align*} $ Multiply the second term by $\dfrac{r + 8}{r + 8}$ $ \begin{align*} \dfrac{4r}{r - 10} \times \dfrac{r + 8}{r + 8} & = \dfrac{(4r)(r + 8)}{(r - 10)(r + 8)} \\ & = \dfrac{4r^2 + 32r}{(r - 10)(r + 8)}\end{align*} $ Now we have: $ = \dfrac{4r^2 - 50r + 100}{(r + 8)(r - 10)} - \dfrac{4r^2 + 32r}{(r - 10)(r + 8)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{4r^2 - 50r + 100 - (4r^2 + 32r)}{(r + 8)(r - 10)} $ $ = \dfrac{4r^2 - 50r + 100 - 4r^2 - 32r}{(r + 8)(r - 10)} $ $ = \dfrac{-82r + 100}{(r + 8)(r - 10)}$ Expand the denominator: $ = \dfrac{-82r + 100}{r^2 - 2r - 80}$